Multi-Fibers Bundles As a New Model for High-Dimensional Spacetimes

Home , Fiber bundle

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In all the paper, (M ,gg) is a semi-Riemannian manifold with metric g, ∇ is its Levi- Civita connection, R and Ric are the Riemann and Ricci curvatures. TpM is the tangent space of M at p. We note ∇∇· T the divergence of a tensor T . In all the paper, we identify, without any comments, a (0, 2)-tensor T with the (2, 0)-tensor T ♯♯ obtained by musical isomorphisms. They represent the same physical object. In particular, the Ein- stein curvature G = Ric − 1/2SS.gg will often be consider as a (2, 0)-tensor, G = Gij. We also consider, for a 2-tensor T , the divergence ∇∇· T as a vector, that is, we identify the 1-form ∇∇· T and (∇∇· T )♯. At last, we shall note eT the endomorphism ﬁeld g-associated e to T : ∀(uu,vv) ∈ Tp(M ) × Tp(M ) : g(uu, T (v)) = T (uu,vv).

1 Beyond ﬁve dimensions, Geometrical setting.

Theoretical evidences, like string theory, suggest the need for a spacetime with more than five dimensions. We want to present in this paper a mathematical structure generalizing the fiber bundle, that enable the possible inclusion of such other dimensions that might model geometrically multiple physical effects. Note however that we do not pretend here to model precisely such other fields, but that we just want to present a possible mathematical frame.

1.1 Multiple fibers at each point of spacetime. The mathematical translation of the heuristic idea of a 4-dimensional classical spacetime equipped with extra "small" dimensions is a fiber bundle structure π : M → M on a (4+ k)-dimensional manifold M , with fiber a compact manifold F of dimension k, more shortly a F -fibration. Heuristically, M is then the "classical" spacetime when we neglect things that happen in the fibers. As we said in the abstract, Kaluza and Klein used a fibration with fiber the standard circle S1, this fiber carrying the electromagnetic potential. Indeed, suppose now that (M ,gg) is a 5-dimensional Lorentz manifold equipped with a S1-fibration π : M → M over a 4-dimensional Lorentz manifold M . Then the electromagnetic potential can be defined as the unit timelike vector field Y tangent at each point x ∈ M to the fiber 1 −1 1 Sx := π (π(x)) passing through x, once some orientation on S has been defined. Details, as well as a new approach to these ideas can be found in our article [ref :A new approach to Kaluza-Klein theory.] Now if we want to keep the result obtained for electromagnetism while including other possible interactions, the fiber F should be of the form F = S1 × W where W is a compact manifold of dimension m and S1 is the classical circle. However, to keep results obtained for electromagnetism in 5 dimensions, through objects naturally given by the action of S1, one faces a important issue : at each point x ∈ M , such a fiber bundle gives a natural fiber F = S1 × W through x, but it does not give a natural fiber through x isomorphic to S1 only ; there is no natural splitting of the fiber S1 × W at x. Therefore, such a fiber bundle alone will not furnish an electromagnetic potential Y . A very elegant extension of the structure of fiber bundle, giving a way to define any 1.2 Splitting of a product manifold. 3 number of natural fibers at each point x of a manifold M , was originally proposed by Michel Vaugon in [ref]: it is the nice structure of observation atlases. It will be exposed in section [?] below in details, but we give here a new approach, based on the more classical notions of fibrations and submersions.

1.2 Splitting of a product manifold. A natural way to split a ﬁber F of the form S × W is based on the following nice construction. Let F , S and W be three compact manifolds and let

Φ=(h, f): F → S × W be a diffeomorphism, where h and f are the components of Φ. Then h : F → S and f : F → W are submersions. We then define unambiguously, for any x ∈ F , two fibers at x by : −1 Sx := f (f(x)) −1 Wx := h (h(x)) Because S and W are compact, a theorem of Ehresmann states that these submersions are in fact fibrations. Using the diffeomorphism Φ, we can see that h and f are more precisely

fibrations with fibers W and S respectively. Indeed, the restrictions h|Sx : Sx → S and f|Wx : Wx → W are diffeomorphisms whose inverse maps are given respectively by :

−1 −1 −1 −1 (h|Sx ) (u) = Φ (u, f(x)). (f|Wx ) (v) = Φ (h(x), v).

Thus :

• h : F → S is a W -ﬁbration.

• f : F → W is a S-ﬁbration.

Furthermore, we have a natural splitting of the manifold F as a product of two ﬁbers at a given point : for a given point p ∈ F , we have a natural diﬀeomorphism:

ψp : F −→ Sp × Wp y 7−→ ( Φ−1(h(y), f(p)) , Φ−1(h(p), f(y)))

Indeed, the inverse map is given by :

−1 F ψp : Sp × Wp −→

(a, b) 7−→ Φ−1(h(a) , f(b))

To prove this, note first that, for (a, b) ∈ Sp × Wp, by definition of the fibers, f(a)= f(p) and h(b) = h(p). Setting y = Φ−1(h(a), f(b)), by definition of Φ and its components h and f, h(y) = h(a) and f(y) = f(b). Therefore, Φ−1(h(y), f(p)) = Φ−1(h(a), f(a)) = a and Φ−1(h(p), f(y))=Φ−1(h(b), f(b)) = b. 1.3 Multi-fiber bundle. 4

1.3 Multi-fiber bundle. We now apply this construction to define a multi-fiber structure on a manifold M .

Let M be a manifold. Let S and W be two compact manifolds, and F := S × W . We say that a manifold M is a multi-fiber bundle with fibers (S, W ), or a (S, W )-fibration, if M is given two smooth maps according to the following diagram

M −→Φ S × W π ↓ M

where π : M → M is a F -ﬁbration, and where Φ=(h, f): M → S × W is a map such that, for any p ∈ M , the restriction F ≃ Φ|Fp : p → S × W

−1 of Φ to the π-ﬁber Fp := π (π(p)) is a diﬀeomorphism. Here again, h and f are the two components of Φ.

−1 −1 We shall note x = π(x) for x in M , and Fx := (S × W )x := π (π(x)) := π (x) the π-fiber at a point x of M . (S × W )x is intuitively {x} × F . We also note U := π(U ) for a set U . For a map f : M → N , where N is a manifold, we sometimes note fx the −1 −1 restriction of f to the π-fiber (S × W )x : fx := f|π (x) := f|π (π(x)) := f|Fp . F F ≃ Using now the splitting of each fiber p with the diffeomorphism Φ|Fp : p → S ×W , as seen in the previous section, we can define three natural fibers at each point p ∈ M :

For p ∈ M , we deﬁne unambiguously three ﬁbers at p :

Let us analyze the requirement made on Φ. As π is a fibration, we have for some neighborhood Ux of any point x, a local trivialization chart φU : Ux → U x × (S × W ) such that p1 ◦ φU = π, where p1 is projection on the first factor. We can then compose φU with projection on S × W to get a function satisfying our requirement on Ux. So what we ask for is the additional requirement that we have in fact a global, well-defined, identification via Φ|Fx of the π-fibers with S × W . 1.3 Multi-fiber bundle. 5

The fundamental idea of multi-fiber structure is that, for a (S×W )-fibration structure on a manifold M , one can furthermore define unambiguously, at each point, objects that depend only on one of the components, S or W , of the global (S × W ) fiber.

The multi-ﬁber structure satisﬁes the following natural and important properties :

′ ′ • Fibers are well deﬁned : p ∈ Sp ⇒ Sp′ = Sp and p ∈ Wp ⇒ Wp′ = Wp.

• Splitting of the ﬁbers : Thanks to the splitting ψp deﬁned in the previous section, ≃ we have at each point p ∈ M a canonical isomorphism : Fp → Sp × Wp

• Adapted charts : Let us fix a point p ∈ M . Using the fibration π, and as all the fibers are diffeomorphic, we have some local trivialization chart

φU : U → U × Fp

for a neighborhood U = Up of p in M . Composing with Id × ψp, where ψp is the splitting of the fiber Fp defined in the previous section, we get an adapted diffeomorphism

ϕU =(Id × ψp) ◦ φU : U → U × Sp × Wp

singularizing the ﬁbers at p. Then, taking coordinates charts on open subsets of U , Sp and Wp respectively, we obtain very useful adapted charts on Up. See below.

• Orientation of the ﬁbers : Let S be given an orientation ; an orientation on W would be treated the same way. For any point y ∈ M , the diﬀeomorphism

hy|Sy : Sy → S can be used to pull-back the orientation on Sy. We say that Φ is compatible with the orientation of S if, in some neighborhood Up of any point p, there is a frame ﬁeld for TySy, y ∈ Up, compatible with the orientation

pulled-back by hy|Sy .

We now consider M to be equipped with a semi-Riemannian metric g. We then deﬁne the horizontal space Hx at a point x as the g-orthogonal space to TxFx in TxM :

⊥ Hx := (TxFx)

We can then consider a compatibility condition between g and the multi-fiber structure: • Signature of the fibers : We say that the metric g is compatible with the multi- fiber bundle structure if the signature of the restriction of g to any fiber as defined above is constant. That is, for any x ∈ M, the signature of g restricted to Sx and Wx is independent of x. In this case, for given signatures σa and σb of adequate length, we will say that g is of signature σa on S and σb on W . The 1.4 Adapted charts. 6

signature of gx on the horizontal space Hx is then also independent of x.

Note that this construction can easily be generalized to define more than 2 fibers at each point of a manifold M . If we are given a global π-fiber of the form F = W1 ×...×Wk, ≃ we essentially replace Φ=(h, f) by Φ=(f1, ..., fk): F → W1 × ... × Wk with adapted analog properties.

1.4 Adapted charts. As they are useful to understand the situation, let us see how we get adapted charts, and what they look like. Consider M with a (S1 ×W )-multi fiber structure. As we saw above, starting with a trivialization chart of the fibration π and composing with the splitting of the fiber Fp by ψp, we have in the neighborhood of any fixed point p ∈ M an adapted diffeomorphism of the following form:

φU 1 Up −→ U p × (S × W )p

Id ↓ ↓ ψp U 1 p × Sp × Wp

1 1 As Sp and Wp are diﬀeomorphic to S and W respectively, we can now take coordinates i U ˙ 1 1 k W (x ) on p, (u) on some neighborhood Sp of p in Sp , and (w ) on some neighborhood p of p in Wp, to obtain a chart of the form :

φU 1 Up −→ U p × (S × W )p

Id ↓ ↓ ψp U ˙ 1 W p × Sp × p ↓ ↓ ↓ (xi, u, wk)

Centering the chart so that the coordinates of p are (0, ..., 0), the coordinates expression of f|Fp and h|Fp are :

1 m 1 m f|Fp (0, ..., 0,u,w , ..., w )=(w , ..., w )

1 m h|Fp (0, ..., 0,u,w , ..., w )=(u) Indeed, they are submersions ! It is then clear that :

−1 ˙ 1 (f|Fp ) (f(x)) = {(0, ..0, u, 0, ..., 0)} := Sp

−1 1 m W (h|Fp ) (h(x)) = {(0, ..., 0, 0,w , ..., w )} := p ˙1 W 1 for some neighborhoods Sp and p of p in Sp and Wp respectively, where the coordinates are deﬁned. 2 Extending Kaluza-Klein model of spacetime. 7 2 Extending Kaluza-Klein model of spacetime.

2.1 Multi-fiber Kaluza-Klein spacetime. We propose as example a possible model for space-time. For illustration of the possibili- ties, and further developments by Michel Vaugon to be found in [ref : New approach to Kaluza-Klein theory, and A Mathematicians’ View of Geometrical Unification of General Relativity and Quantum Physics], we choose the fifth dimension to be timelike. The reader uncomfortable with the two timelike dimensions, can still consider the signature on S1 to be spacelike, all what follows will remain essentially unchanged.

Let S1 be the classical circle with a chosen orientation, and let W be a compact manifold of dimension m. Spacetime is a semi-Riemannian manifold (M ,gg) of dimension 5+ m equipped with a multi-ﬁber bundle structure, of ﬁber F = S1 × W :

M −→Φ F = S1 × W π ↓ M

as deﬁned above. We suppose that the metric g is compatible with the multi-ﬁber bundle structure, g being of signature (−1) on S1, and (+, ..., +) on W ; the total signature of g is thus (−, +, +, +, −, +, ..., +). M is "classical" spacetime. 1 ⊥ The horizontal space at a point x is Hx := (Tx(S ×W )x) , and gx is of signature (−, +, +, +) on Hx. The horizontal space Hx is naturally isometric to Tπ(x)M ; it represents the local and classical Minkowski spacetime at x.

The eﬀect of the "extra" m dimensions carried by W will be modeled via the geometry of the maps π : M → M and Φ: M → S1 × W , and via the metric g or its Einstein curvature G. The idea is also that what passes to the quotient can be neglected.

We can also add to the definition of g being compatible with the given (S1, W )-multi- fiber structure on M the following requirement : for any pair of adapted charts φi, φj U U ∗ ∗ as defined above, ∀x ∈ i ∩ j, φi (∂t)x and φj (∂t)x are timelike and in the same time ∗ ∗ orientation, i.e. g(φi (∂t)x,φj (∂t)x) < 0, where ∂t is the tangent vector to the canonical 4 coordinates (t, x, y, z) on Θi ⊂ R . This condition gives a "classical" time-orientation on every apparent space-time Hx, varying differentially with x.

The Electromagnetic potential. We suppose that we have the canonical standard orientation on S1. We can then define: We define Y to be the vector field defined at each x ∈ M to be tangent to the fiber 1 1 Sx and such that g(YY,YY ) = −1, with the chosen orientation for S ; Y is called the electromagnetic potential. We then define the differential F := d(Y ♭) where Y ♭ 2.2 Application to an electrically charged fluid 8

is the 1-form associated to Y by g ; F is the electromagnetic field. We always suppose from now on that Y is a Killing vector field. (Note that if Y is Killing and of constant norm, it is necessarily geodesic.) The local diffeomorphisms generated by Y are therefore isometries.

The next proposition, easy to prove, shows why it is natural to suppose that Y is a Killing vector ﬁeld, when supposing that the compact dimensions are "small":

1 1 Proposition 1. Averaging the metric on S : Let Y be tangent to the fiber Sx and such that g(Y,Y )= −1 as above, but without supposing that Y is a Killing vector field. Let σ be the 1-parameter group of diffeomorphisms associated to the flow of Y . Define the "averaged" metric g by :

t0+ℓx M 1 ∗ ∀x ∈ , gx := (σ (t)g)x.dt ℓx Zt0

1 where ℓx is the length of Sx relative to g. (gx does not depend on the choice of t0 as ∗ σx(.) is periodic, of period ℓx.) Then, g(YY,YY )= −1, and ∀s ∈ R, σ (s).g = g. That is, Y is a Killing vector ﬁeld for g.

2.2 Application to an electrically charged fluid To illustrate possible use of our multi-fibers structure, we give an application to a new approach to Kaluza-Klein theory. Details and proofs can be found in [3]. We start by considering the inclusion of electromagnetism in the classical frame of General Relativity. Then we show how the multi-fibers structure gives a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy. 1 e We note G = Ric − 2SS.gg the Einstein curvature. G is the associated endomorphisms e field. We then define GH , the endomorphisms field on the horizontal subspaces Hx, de- e e fined by GH = prH ◦ ( G|H ), where for x ∈ M , (prH )|x is the orthogonal projection of TxM on Hx. This tensor will be very important to define fluids in our extended Kaluza- Klein spacetime.

Dust charged matter ﬂuid

Deﬁnition 1. A domain Ω ⊂ M is a dust charged matter ﬂuid domain if and only if its Einstein curvature tensor can be written at every point

G = µX ⊗ X + αYY ⊗ Y

with the condition that, at each point x, prH (X) is a basis for a timelike 1-dimensional e eigenspace of GH of eigenvalue −µ< 0. X is then unique for this decomposition. 2.2 Application to an electrically charged ﬂuid 9

Associated "classical" data : For such a perfect ﬂuid without pressure, there is a unique decomposition (once a time orientation is chosen) :

G = µX0 ⊗ X0 + e(X0 ⊗ Y + Y ⊗ X0)+ γYY ⊗ YY,

where g(X0,X0) = −1 and X0⊥Y . µ is called mass density, e the charge density. These are canonically given by :

µ = −G(X0,X0)

e = G(YY,X0) γ = G(YY,YY ) We then have e X = X + Y 0 µ e The vector field X = X0 + µY is called the vector field of the fluid, and the associated flow, the flow of the fluid. The vector field X0 will be called the apparent, or visible, field of the fluid, and the associated flow, the apparent, or visible, flow. Note that at each point, by definition, X0(x) ∈ Hx.

The following theorem is a consequence of the purely geometric Bianchi identity ap- plied to the Einstein curvature G. No "laws" need to be added.

Theorem 1. Dynamics of charged dust. For the domain of a perfect charged ﬂuid without pressure where G and its associated classical data are written :

G = µX ⊗ X + αYY ⊗ Y = µX0 ⊗ X0 + e(X0 ⊗ Y + Y ⊗ X0)+ γYY ⊗ YY,

Bianchi identity gives:

e • Conservation Laws: X0( µ )= ∇∇· (µX0)= ∇∇· (eX0)=0

♯ • Maxwell equations : dF =0 and (∇∇· F ) =2eX0 − (2γ + S g)Y

e • Free Fall : X = X0 + µY is a geodesic vector ﬁeld.

e • Lorentz equation: µ∇X0 X0 = e. F (X0). This is just free fall read on H. When projected on the "classical" 4-dimensional space-time H = Y ⊥, these equations are the classical equations of physics. e Note that Lorentz equation is obtained from the geodesic motion of X = X0 + µY by developing ∇X X =0 and writing (projecting) this equation on the horizontal space ⊥ H = Y , noting that ∇X0 X0⊥Y , which means that ∇X0 X0 ∈ H. We therefore see that : 2.2 Application to an electrically charged ﬂuid 10

Free fall for X is equivalent to Lorentz equation for X0.

(Remember that the ﬁrst Maxwell equation, dF = 0, is always obvious as we set F = dY ♭.) Proof. The proof is available in [3], but here is the scheme : First some properties of Y are established. Then, considering ∇∇· G as a vector, i.e. identifying ∇∇· G and (∇∇· G)♯, and noticing that ∇∇· G = 0 by Bianchi identity, one compute:

• g(∇∇· GG,YY ), this will be charge conservation law.

• g(∇∇· GG,X0), this will be mass (baryonic number) conservation law.

• prH (∇∇· G), this be the equation of motion. In the case of dust prH (∇∇· G)= ∇∇· G, so the equation of motion is simply ∇∇· G =0.

Then, ∇∇· F = ∇∇· (dY ♭) is computed, which gives the second Maxwell law, the ﬁrst, dF =0, being obvious as F = d(Y ♭). Finally, to prove that the ﬂow of X is geodesic, we simply compute ∇X X, noticing e that X0( µ )=0 ; this leads to ∇X X =0.

It is very important to note that we have never mentioned any kind of energy- momentum tensor. The point is that from our point of view, this concept has no meaning. Indeed, here is our frame of ideas : A/: Space-time is a five dimensional semi-Riemannian manifold satisfying definition 1 B/: Instead of defining an energy-momentum tensor, we caracterize a domain of spacetime by a geometric type. We then define physical concepts by geometric caracteristics of curvature. C/: Physical equations are projection of the Bianchi identity ∇∇· G = 0 on the 4- dimensional subspace H = Y ⊥ modelizing our classical 4-dimensional space-time.

General charged matter fluid In the result above, we only used an object, Y defined on the S1 component of the total fiber F = S1 × W . Now, with the same notations as for a dust fluid, we can define a general fluid to be a domain D of M where the Einstein curvature can be written :

e2 G = µX ⊗ X + αYY ⊗ Y + P = µX ⊗ X + e(X ⊗ Y + Y ⊗ X )+(α + ).YY ⊗ Y + P. 0 0 0 0 µ

Here, the tensor P is the pressure, or constraint, tensor of the ﬂuid. If we note Tx the

subspace of dimension 2 of TxM generated by X0(x) and Yx, then P is just P = G −G|Tx ,

where G|Tx = µX ⊗ X + αYY ⊗Y . We then deﬁne the apparent pressure Pv as the pressure P restricted to the horizontal space Hx ; and the hidden pressure as Ph := P − Pv. 3 A far-reaching example : a (S1 × S3)-multi-ﬁber structure on M . 11

The ﬂuid will be called perfect if Ph(Y )=0. In matrix form, with suitable basis for 1 TxH, TxSx and TxWx, and with some abuse of notation for Ph, G can then be written: µ 0 0 0 e  0 0  0 P 0 P G =  v h   0 0     e 0 0 0 γ 0     P 0 P   h h  This will now be our general model for a ﬂuid. Then, an analog proof to the above theorem gives :

Theorem 2. Equations for the spacetime dynamics of ﬂuids. If D is a ﬂuid domain as above, Bianchi identity gives :

• Energy Conservation Laws : ∇∇· (µX)= ∇∇· (µX0)= hX0,∇∇· P i

e • Electric Charge Conservation Law : ∇∇· (eX) = ∇∇· (eX0) = ∇∇· ( P (Y )) = hY,∇∇· P i

• Motion Equations :

– For the ﬂuid : µ∇X X = −∇∇∇· P −hX0,∇∇· P iX

e ⊥ – For the apparent ﬂuid : µ∇X0 X0 = e. F (X0) − prT (∇∇· P )

e • Maxwell Equations : dF =0, and ∇∇· F = e.X0 +1/2|F |g.Y − P (Y )

Our first theorem above is therefore a special case where the pressure P =0. In this last theorem, the pressure P can be split everywhere into P = Pv+Ph, to show physical effects due to the three "classical" dimensions, Pv, and those due to the extra "hidden or small" dimensions, Ph. This theorem therefore shows that Ph, the hidden pressure, is a possible way to model or encode deviations from standard 4-dimensional General Relativity, or "dark" effects such as dark matter or energy. Indeed, when Ph =0, we recover classical equations for a perfect fluid with pressure.

Full details of this use of the multi-ﬁbers structure can be found in [3] : S. Collion, M. Vaugon, A New Approach to Kaluza-Klein theory. arXiv.

3 A far-reaching example : a (S1 ×S3)-multi-ﬁber structure on M .

The 3-dimensional sphere S3 is the classical geometric space used in quantum theory to describe spin. Indeed, S3 carries a natural frame ﬁeld as well as canonical endomorphisms 3 A far-reaching example : a (S1 × S3)-multi-ﬁber structure on M . 12

giving precisely the spin matrixes. This lead the second author of this article to use a multi-fiber bundle structure on a manifold M with fiber F = S1 × S3 to give a geometric model of electromagnetic and spin effects on M . The S1 component was used to define the electromagnetic potential, and the S3 component to define objects related to the spin. Furthermore, the modelization of the physical effects of both electromagnetism and spin, for instance to describe the Stern and Gerlach experiment, required to use the geometry of the total fiber F = S1 × S3, so the full multi-fiber structure, given by the two maps π and Φ of the definition, was used. More precisely, this S1 × S3-multi-fibers structure, and the objects described below that can be built with it, was the geometric starting point for the second author to build a new approach towards a geometric unification of General Relativity and Quantum Physics. It is based on special metrics on a (5 + k)-dimensional manifold modelizing quantum particles physics in spacetime. In this setting, only the metric is relevant, no objects or laws are added, these appear as geometric quantities issued from curvature and geometric theorems, such as Bianchi identity, linked with the objects described below. See [19]. Here, we just give some examples showing how the multi-fiber structure can be used to define various objects, linked to the choice of multiple fibers, at each point of the manifold.

We therefore go over the problematic once more : if we want to add to our geometric inclusion of electromagnetism in the frame of General Relativity a possible modelization of spin, we need a way to keep the definition of the electromagnetic potential Y . With just a classical (S1 × W )-fiber bundle structure on M , as we said, they is no well defined 1 M 1 fiber Sx trough x ∈ . However, by considering the (S , W )-multi-fibers structure on M 1 M , there is a naturally defined Sx trough each x ∈ , so we can expect to add other physical effect on M , by choosing for example here W = S3, while keeping objects defined with the first component, S1, of the global fiber F = S1 × W . This is indeed the main purpose of the multi-fiber bundle structure.

We therefore consider that the spacetime (M ,gg) is a (S1,S3)-multi-ﬁbers bundle :

M −→Φ F = S1 × S3 π ↓ M

M 1 3 At each point x ∈ , we then have two naturally defined fibers : Sx and Sx. As before, S1 gives the electromagnetic potential Y : Y is the vector field defined at M 1 each x ∈ to be tangent to the fiber Sx and such that g(Y,Y ) = −1, with the chosen orientation for S1. 3 Now, S is a parallelizable manifold. Therefore, there exist three vector fields X1,X2,X3 3 3 3 on S giving at each point p ∈ S a basis of the tangent space TpS . Let us fix such a sphere 3 M 3 3 3 S with its frame field. If we use at each point x ∈ the diffeomorphism f|Sx : Sx → S as seen in the definition of the multi-fibers bundle, we define unambiguously three vector 4 Observation Atlases. 13

3 −1 3 fields Xj by Xj(x) := (Tf|Sx ) (Xj(f(x)), where Tf|Sx is the differential (tangent map) 3 of f|Sx at x. f f i We can do more. For a point x ∈ M , let ∆x be the geometric Laplacian (∆= −∇ ∇i) 3 3 of the Riemannian manifold (Sx,gg|Sx ). We then have the spectral set {λ0(x),λ1(x), ...} of eigenvalues of ∆x. This define smooth functions x 7→ λk(x) on M . None of these objects could be defined with a simple (S1 × S3)-fiber bundle structure on M .

4 Observation Atlases.

We now present another and equivalent approach to the multi-ﬁber bundle structure. Let be given a manifold M of dimension n = p + m, and a compact manifold W of dimension m. Consider a domain D of M.

Definition 2. An observation diffeomorphism on D relative to W , or for short, a W -chart on M is a triple (V, φ, Θ) where V is an open subset of D, Θ an open subset of Rp, and φ a diffeomorphism from V onto Θ × W .

Deﬁnition 3. A (observation) W -atlas on D ⊂ M is a family ((Vi,φi, Θi))i∈I of W -charts such that :

• i∈I Vi = D S −1 1 −1 1 1 • ∀i, j ∈ I, ∀x ∈Vi ∩Vj, φi ({φi (x)} × W )= φj ({φj (x)} × W ), where φi is the component function of φ going to Θi.

It is quite easy to prove that this deﬁnition is in fact equivalent to that of a ﬁbre bundle :

Theorem 3. With the notations above, the two following propositions are equivalent:

• There exists an observation W -atlas on D.

• There exist a smooth manifold B of dimension n−m and a submersion π : D → B such that (D,π,B) is a ﬁbre bundle with ﬁbre W .

Remark: the compacity of W is required to prove that B is Hausdorf. If not, one −1 1 needs to require in deﬁnition 5 that any φi ({φi (x)} × W ) is closed in D.

We now want to consider the case where W is a product manifold (for example W = 1 ′ S × W ). We therefore consider two compact manifolds Wa and Wb. 4 Observation Atlases. 14

Deﬁnition 4. For manifolds D and W = Wa × Wb as above, an observation Wa-atlas is a family ((Vi,φi, Θi))i∈I of W -charts such that :

• i∈I Vi = D S • ∀i, j ∈ I, ∀x ∈Vi ∩Vj,

−1 1 b −1 1 b φi ({φi (x)} × Wa ×{φi (x)})= φj ({φj (x)} × Wa ×{φj(x)})

1 b where φi is the component function of φ going to Θi and φi the component going to Wb.

We deﬁne in the same way a Wb-atlas. If Wa is oriented, we say that the Wa-atlas preserves the orientation of Wa if −1 1 b ∀i, j ∈ I, ∀x ∈Vi ∩Vj, the orientation on φi ({φi (x)} × Wa ×{φi (x)}) carried from Wa by φi is the same as that carried by φj. (One write the same deﬁnition if Wb is oriented).

Deﬁnition 5. An observation (Wa, Wb)-atlas on D is a family ((Vi,φi, Θi))i∈I of W - charts giving both a Wa-atlas and a Wb-atlas.

Caution : a (Wa, Wb)-atlas is not equivalent to a (Wa × Wb)-atlas.

Definition 6. If A is a (Wa, Wb)-atlas, for every x ∈ D and any W -chart (Vi,φi) around x, we can define, naturally relatively to A, fibers through x by

−1 1 b Wa,x := φi ({φi (x)} × Wa ×{φi (x)})

and −1 1 a Wb,x := φi ({φi (x)}×{φi (x)} × Wb)

which are respectively isomorphic to Wa and Wb and, this is the important point, these ﬁbers do not depend on the chosen W -chart. If Wa is oriented, so are all the ﬁbers Wa,x. ′ Note the important fact that if x ∈ Wa,x, then Wa,x′ = Wa,x.

Caution : With a (Wa, Wb)-atlas, not being a (Wa × Wb)-atlas, there is no naturally deﬁned ﬁber Wx isomorphic to Wa × Wb.

The link with the multi-ﬁber structure is given by the following proposition : REFERENCES 15

Theorem 4. With the notations above, the two following propositions are equivalent:

• There exists an observation (Wa × Wb)-atlas on D that is also a (Wa, Wb)-atlas. • There exist a smooth manifold B, a submersion π : D → B and a map Φ = (fa, fb): F → Wa × Wb such that

Φ D −→ F = Wa × Wb π ↓ B

is a multi-ﬁbers bundle with ﬁbers (Wa, Wb).

Given a manifold M equipped with a (Wa, Wb)-altlas, a metric g is compatible with the atlas if the signature of the restriction of g to any ﬁber as deﬁned above is constant. That is, for any x ∈ M, the signature of g restricted to Wa,x and Wb,x is independent of x. In this case, for given signatures σa and σb of adequate length, we will say that g is of signature σa on Wa and σb on Wb. The signature of gx on the subspace of TxM orthogonal to Tx(Wa,x ×Wb,x) is then also independent of x; we call this subspace the horizontal space at x.

There is a natural decomposition TxM = Hx⊥TxWa,x ⊕ TxWb,x. Note that TxWa,x is not necassarily orthogonal to TxWb,x. Hx will be called the apparent space-time at x. Caution : the ﬁeld of tangent space Hx need not be integrable ; there is not in general, even locally, a 4-dimensional submanifold of M tangent at every x to H. To conclude this section, we recall basic deﬁnitions concerning complete atlases.

1/: Two observation (Wa, Wb)-atlases, possibly each compatible with a metric g, are said to be equivalent, it their union is still an observation (Wa, Wb)-atlas, compatible with g. 2/: The completed (Wa, Wb)-atlas associated to a (Wa, Wb)-atlas A, is the union of all the (Wa, Wb)-atlases compatible with A. 3/: A (Wa, Wb)-atlas is complete if it is equal to its completed (Wa, Wb)-atlas.

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